2D Electron Transport Across the Ballistic-Hydrodynamic Crossover in Complex Geometries

Collective motion of two-dimensional electrons exhibits a rich crossover between ballistic and hydrodynamic flow when the electron–electron (e–e) scattering time becomes comparable to device and drive scales. I present a kinetic theory framework based on an angular-harmonic expansion of the Boltzmann equation, with the simplified BGK approximation of the Fermionic collision integral, that systematically interpolates across regimes, retaining enough harmonics to capture both viscous and long-lived modes responsible for nonlocal and tomographic dynamics. The approach reproduces known hydrodynamic signatures such Poiseuille flow in appropriate limits and quantitatively connects them to boundary conditions and sample geometry. 

We apply our scheme to a variety of complex two-dimensional geometries relevant to experiment to showcase the applicability of our methods. The resulting phase diagrams in the momentum relaxing and momentum conserving scattering times, τ_mr and τ_mc , delineate the three regimes of transport, providing robust predictions for streamlines, vorticity, and nonlocal voltage patterns accessible to scanning probe and multi-terminal measurements. Time-dependent (AC) response is also accessible for our real-time algorithm.  Our framework thus offers a unified, systematically improvable route to modeling electron flows in high-mobility 2D materials that also readily generalizes to nonlinear transport near the hydrodynamic limit.